The signed distance-$k$-domination number of a graph is a certain variant of the signed domination number. If $v$ is a vertex of a graph $G$, the open $k$-neighborhood of $v$, denoted by $N_k(v)$, is the set $N_k(v)=\lbrace u\mid u\ne v$ and $d(u,v)\le k\rbrace $. $N_k[v]=N_k(v)\cup \lbrace v\rbrace $ is the closed $k$-neighborhood of $v$. A function $f\: V\rightarrow \lbrace -1,1\rbrace $ is a signed distance-$k$-dominating function of $G$, if for every vertex $v\in V$, $f(N_k[v])=\sum _{u\in N_k[v]}f(u)\ge 1$. The signed distance-$k$-domination number, denoted by $\gamma _{k,s}(G)$, is the minimum weight of a signed distance-$k$-dominating function on $G$. The values of $\gamma _{2,s}(G)$ are found for graphs with small diameter, paths, circuits. At the end it is proved that $\gamma _{2,s}(T)$ is not bounded from below in general for any tree $T$.
The concept of signed domination number of an undirected graph (introduced by J. E. Dunbar, S. T. Hedetniemi, M. A. Henning and P. J. Slater) is transferred to directed graphs. Exact values are found for particular types of tournaments. It is proved that for digraphs with a directed Hamiltonian cycle the signed domination number may be arbitrarily small.